0,2

Least side in (m,m+1,m+2) integer-sided triangle with integer area.

Also describes triangles whose sides are consecutive integers and in which an inscribed circle has an integer radius - Harvey P. Dale (hpd1(AT)is2.nyu.edu), Dec 28 2000

Equivalently, positive integers m such that (3/16)*m^4 + (3/4)*m^3 + (3/8)*m^2 - (3/4)*m - 9/16 is a square (A000290), a direct result of Heron's formula. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 04 2005

a(n) = 3 + floor((2+sqrt(3))*a(n-1)), n>=3. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 04 2005

a(n) = 4*a(n-1) - a(n-2) + 2.

G.f.: (1-2x+3x^3)/((1-x)(1-4x+x^2))=(1-2x+3x^2)/(1-5x+5x^2-x^3); a(n)=(2+sqrt(3))^n+(2-sqrt(3))^n-1; a(n)=2*A001075(n)-1. - Paul Barry (pbarry(AT)wit.ie), Feb 17 2004

Equals A003500(n)-1. - T. D. Noe (noe(AT)sspectra.com), Jun 17 2004

Ralf Stephan's formula (shown in his program) is the simplest (if proved) not involving a recurrence or a different sequence.

(PARI) for(a=1, 10^9, b=a+1; c=a+2; s=(a+b+c)/2; if(issquare(s*(s-a)*(s-b)*(s-c)), print1(a, ", "))) (Shepherd)

(PARI) a(n)=if(n<1, 1, -1+ceil((2+sqrt(3))^(n))) (from R. Stephan)

Corresponding areas are in A011945.

Cf. A001353, A019973 (2+sqrt(3)), A102341, A103974, A103975.

Sequence in context: A008827 A026529 A101052 this_sequence A163774 A014985 A015521

Adjacent sequences: A016061 A016062 A016063 this_sequence A016065 A016066 A016067

nonn

Robert G. Wilson v (rgwv(AT)rgwv.com)

More terms from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Feb 18 2007